Core Idea: If the integrand has the form f(g(x))*g'(x), substitute u = g(x) to simplify.

Pattern 1: Linear Substitution integral f(ax+b) dx = (1/a)F(ax+b) + C Example: integral cos(3x+2) dx = (1/3)sin(3x+2) + C

Pattern 2: Function and its Derivative integral [f(x)]^n * f'(x) dx = [f(x)]^(n+1)/(n+1) + C Example: integral sin^5(x)*cos(x) dx = sin^6(x)/6 + C (u = sin(x))

Pattern 3: Reciprocal with Derivative integral f'(x)/f(x) dx = ln|f(x)| + C Example: integral (2x+1)/(x^2+x+3) dx = ln|x^2+x+3| + C

Pattern 4: Adjusting Constants When the numerator is close to but not exactly the derivative: integral (4x+7)/(2x^2+7x+3) dx: Note d/dx(2x^2+7x+3) = 4x+7. Perfect match! = ln|2x^2+7x+3| + C

When it's not a perfect match, split: integral (3x+5)/(x^2+4x+1) dx: d/dx(x^2+4x+1) = 2x+4. Write 3x+5 = (3/2)(2x+4) + 5-6 = (3/2)(2x+4) - 1. Split into two integrals.

Pattern 5: Trigonometric Substitutions for Radicals

• sqrt(a^2-x^2): x = asin(t), dx = acos(t)dt, sqrt becomes a*cos(t)
• sqrt(a^2+x^2): x = atan(t), dx = asec^2(t)dt, sqrt becomes a*sec(t)
• sqrt(x^2-a^2): x = asec(t), dx = asec(t)tan(t)dt, sqrt becomes atan(t)

Pattern 6: Reciprocal Substitution For integrals with 1/x^n terms: let x = 1/t, dx = -dt/t^2. Useful for integral dx/(x^2*sqrt(x^2-1)) type problems.